# Fourier coefficients of half-integral weight modular forms in arithmetic progressions

## TL;DR

This paper investigates the distribution of Fourier coefficients of half-integral weight modular forms in arithmetic progressions, revealing they follow a mixed Gaussian distribution similar to Salié sums, extending previous results to prime power moduli.

## Contribution

It extends the probabilistic analysis of Fourier coefficients to half-integral weight forms and prime power moduli, demonstrating a mixed Gaussian distribution in this context.

## Key findings

- Fourier coefficient sums follow a mixed Gaussian distribution.
- Results extend previous work to half-integral weight forms.
- Distribution relates to asymptotic behavior of Salié sums.

## Abstract

We study the probabilistic behavior of sums of Fourier coefficients in arithmetic progressions. We prove a result analogous to previous work of Fouvry-Ganguly-Kowalski-Michel and Kowalski-Ricotta in the context of half-integral weight holomorphic cusp forms and for prime power modulus. We actually show that these sums follow in a suitable range a mixed Gaussian distribution which comes from the asymptotic mixed distribution of Sali\'e sums.